保守振动方程周期解的存在性研究

项目名称： 保守振动方程周期解的存在性研究

项目编号： No.11271364

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 刘文斌

作者单位： 中国矿业大学

项目金额： 57万元

中文摘要： 本项目研究保守振动方程周期解的存在性，研究高阶、高维微分方程周期问题Fucik谱的结构和分数阶微分方程周期问题谱的结构。在振动分析中，利用振动的谱点和Fucik谱点，控制势能函数的增长阶和渐进状态，给出在跨共振点情况下保守振动方程周期解存在性的若干判据；在谱和Fucik谱的框架下，探求保守型微分方程、p-Laplacian微分方程以及分数阶微分方程周期解存在性之间的内在关系，以期揭示保守型振动方程周期解存在性的机理和统一规律；建立高阶、高维微分方程和分数阶微分方程周期解存在的有效判据；对所研究的问题给出数值模拟，并对相关的保守系统和频谱不对称的时变保守系统的动力学行为进行实证分析。项目所得结果将改进、丰富和完善已有的相关工作，为工程技术、经济分析等相关问题提供理论支撑。

中文关键词： 微分方程；周期解问题；边值问题；解的存在性；Fucik谱

英文摘要： In this project, we will study the existence of periodic solutions for conservative oscillatory equations, the structure of Fucik spectrum for higher order and multi-dimension equations, and the structure for fractional differential equations under periodic conditions. In oscillation analysis, by using oscillatory spectrum and Fucik spectrum to control increasing order and asymptotic behavior for potential function, we will yield the criteria of existence of periodic solutions for conservative equations under crossing resonance points; seek interrelation of existence of periodic solutions among conservative types of differential equations, p-Laplacian equations and fractional differential equations under the frame of the spectrum and Fucik spectrum, so that reveal mechanism and uniform law of existence of periodic solutions; establish criteria of existence of periodic solutions for higher order and multi-dimension differential equations and fractional differential equations; take numerical analysis for studied problems, and take empirical analysis to dynamic behavior of the relative conservative systems and the time-varying conservative systems of asymmetrical frequency spectrum.The results obtained this project will improve、enrich and complete the known relative works and contribute theoretic support to enginee

英文关键词： Differential equation；Problem of periodic solution；Boundary value problem；Existence of solution；Fucik spectrum